Problem 39 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. $$ \frac{21}{49} $$

Step-by-Step Solution

Answer

The simplest form of the fraction \( \frac{21}{49} \) is \( \frac{3}{7} \).

Step 1: Determine the Greatest Common Divisor

The first step is to find the greatest common divisor (GCD) of 21 and 49, which is the largest number that both 21 and 49 can be divided by evenly. In this case, the GCD of 21 and 49 is 7.

Step 2: Simplify the fraction

Next, divide both the numerator and the denominator of the fraction by the GCD. So, \( \frac{21}{7} = 3 \) and \( \frac{49}{7} = 7 \). Therefore, the simplified version of the fraction is \( \frac{3}{7} \).

Key Concepts

Understanding the Greatest Common Divisor

The concept of the Greatest Common Divisor, or GCD, is essential in simplifying fractions. It's the largest positive number that evenly divides two or more integers without leaving a remainder. In the context of fractions, the GCD helps us identify how much the numerator and denominator can be divided by to reduce the fraction to its simplest form.
The process usually involves a bit of observation and sometimes, trial and error. For two numbers like 21 and 49, you observe their factors:

• The factors of 21 are 1, 3, 7, and 21.
• The factors of 49 are 1, 7, and 49.

The greatest common factor they share is 7, so the GCD is 7. Once you find this number, it'll act as a tool to reduce both the numerator and denominator.

Numerator and Denominator Explained

In any fraction, two important numbers come into play: the numerator and the denominator. The numerator is the top number and the denominator is the bottom number. These two numbers together form a ratio or fraction of a whole.
For example, in the fraction \( \frac{21}{49} \), 21 is the numerator and 49 is the denominator. The numerator represents how many parts you have, while the denominator shows into how many equal parts the whole is divided.
When simplifying a fraction, both the numerator and denominator will change, but crucially, their ratio remains equivalent. This balance is key to maintaining the value of the fraction while reducing its complexity.

Steps to Fraction Reduction

Fraction reduction, also known as simplifying a fraction, involves making the fraction as simple as possible. This means reducing it such that the numerator and denominator are as small as they can be while still keeping the fraction equivalent to the original.
The process for reducing a fraction is straightforward:

• Identify the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.

Using our example, the GCD of 21 and 49 is 7. By dividing both the numerator (21) and the denominator (49) by 7, we reduce the fraction \( \frac{21}{49} \) to \( \frac{3}{7} \).
This simplified form is easier to understand, use, and compare to other fractions, making it a critical step in many math problems.